Show that the two definitions of continuity in section 2.1 are equivalent. Consider separately the cases...

Show that the two definitions of continuity in section 2.1 are equivalent. Consider separately

the cases where z0 is an accumulation point of G and where z0 is an isolated point of G.

2.1 :

Definition1. Suppose f : G → C. If z0 ∈ G and either z0 is an isolated point of G or lim f(z) = f(z0) (z→z0)
then f is continuous at z0. More generally, f is continuous on E ⊆ G if f is continuous at every z ∈ E.

Definition 2.

Suppose f : G → C and z0 ∈ G. Then f is continuous at z0 if, for every positive real

number ε there is a positive real number δ so that
|f(z)−f(z0)|<ε for all z∈G satisfying |z−z0|<δ.

Thanks.

Expert Solution

Colby Messinger answered 1 year ago

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